A few months back I came across an example of a Common Core math problem that really set me off. I mean, it actually offended me. This was the key question for a 6-year-old’s math exam:
(If you’ve never seen this problem before, feel free to try to work it out before reading the Answer after the “read more” link below.)
Recently I came across a year-old article in the Washington Post where the author linked back to the same test again, and the feelings of ire swelled within me again.
Because in addition to the convoluted, logical insanity, it’s a math question that has the math wrong.
The “Answer” to the Problem
According to the conventional wisdom (and what I had to be told myself), the answer to the Common Core question above is “D. 1”. Evidently it is supposed to be assumed that the coins are to be scooped up and placed inside the cup, and the difference between the number of coins you put in, and the number that should be in the cup (“6”), is 1.
Not only does this require a number of illogical and arbitrary assumptions (more on this below), this is also an incorrect conclusion given the clues provided.
Understanding Numbers – Nominal Numbers
In order to understand why the basic premise is so far out of whack, we have to define some terms. In this case, the terms have to do with understanding what numbers mean.
Nominal numbers. At the basic level, a nominal number is like a label. Jackie Robinson wore the number ‘42’ on his jersey, but that was a means of identifying him on the ball field. He wasn’t 42nd in line, nor was he one player worse than whoever wore 41. Labels are effectively useless for mathematical operations because they have no inherent value.
What’s Wrong with the Cup
The example above, the cup has the nominal number of ‘6.’ With no other indication of what is to be determined, it is very clearly a label on the cup with the identifier ‘6.’
In other words, to a six-year old, it looks something like this:
The identity of an item is a non-mathematical construct. That is, you cannot apply a mathematical operation to an identity, only the value of an identity.
Understanding Numbers – Ordinal and Cardinal Numbers
A Cardinal number is what we usually think of when we count things. How many coins do we have in the photo below?
There are 9 coins in the photo above. Note, we are simply counting coins, not evaluating their value. I use this photo to illustrate the point that we are only counting the number of coins, not evaluating their worth. If we want to do that, things get a bit difficult because none of the coins use the same denomination of value.
What about these coins?
Here we have two rows of five coins each. On the top row, take a look at the 4th coin from the left. It is a 5-cent nickel. But note that its value is not related to the 3rd position. In other words, we have not ranked these coins according to value. This number – 4th – is called a ordinal number. Note: We could rank them in order of value, but the number “4” in this case would still be a ordinal value, because the 4th position is not tied to the identity of the coin in that position. In other words, with ordinal numbers we look at the position of the item, not it’s relational value to what’s next to it.
In the test question, it “seems” like we are talking about Cardinal numbers with the coins. That is, we can count that there are 5 coins. This is the “part I know,” according to the test.
Or is it?
The labels underneath the drawings really cause a great deal of confusion here. Take a closer look at the cup:
The cup very clearly already has something in it. There are no coins in that cup that are visible to the test-taker. Yet, this is considered to be ‘whole.’ There doesn’t really appear to be any room for anything else in there.
Oh, wait, I see. I’m supposed to assume some magical connection between coins and a cup (without using Tarot cards), but I’m not supposed to look at the entire drawing. Just the magic number on the cup.
So what’s in the cup, unicorn tears?
Screw you, Common Core.
Am I being pedantic? You bet. The creators of the question have demanded that I make not only an assumption about the relationship between the coins and the cup, but also do not assume too much. That is, assume that I’m supposed to put the coins in the cup, but not assume that there is anything already in the cup even though there clearly is!
Understanding Numbers – Interval Numbers
Okay, so maybe the coins in the exercise are the same. Maybe it’s like these coins:
In the case of these kinds of numbers, each of the coins has the same numerical value, so that the interval between them indicates the amount of value from the total. In this example, each coin is worth $.25, and the 3rd coin is worth 25 cents more than the previous coin. In this case, you can count the number of coins and make an assumption about their value based upon the number of coins and their position (moving left-to-right, for example).
Now, looking at the test question, they kind of look like 25-cent quarters, don’t they? I mean, when you look at the size of the coins next to the cup, they seem like the right size, no? Or maybe they’re $.50 pieces?
Well, if that’s the case, then I’ve got $1.25 in quarters in the exercise, and somehow I’m supposed to have it make up $6. Maybe. Well, that means that I need to come up with $3.75, or 15 more quarters (or, if they’re $.50 pieces, I need to come up with 5 more of them), to make my $6 – assuming that I’m supposed to figure out the value of the coins.
Crap, that can’t be it. I don’t have a “15” or a “5” as a choice. Thank god there was no “5” as a possibility! I would have gotten the answer right! Or wrong. Or.. or crap. I don’t know what I mean.
After all, if you’re going to play fast and loose with assumptions, why not use the value of the coins?
Because the question didn’t mention the coins’ value.
No kidding, Sherlock. It also doesn’t mention what the hell the cup is supposed to represent either. Or what’s in the cup. Or how high the cup is – hey, there’s an idea! Maybe “6” is the height of the cup in coins! Maybe if I stack up the coins, I can see how many it takes to get to the cup’s height!
Hey, why not?
Using the Same Logic
Using the Common Core’s logic, let’s take a look at the exact same parallel example:
So, knowing what we know about the nature of numbers, it’s a bit easier to see where the confusion comes into play.
If these race cars have numbers on them (nominal), does this mean that #51 fell into the sinkhole?
Or, using the logic of the test question, we know there are 6 race cars. Does this mean that there should be 45 cars to add up to the number of cars that can fit in the whole hole?
Oh really? So is that car #51 (nominal), or the 51st car (ordinal), or the last of 51 (cardinal) cars in the hole? According to the assumptions we make in order to create the rules of engagement for answering the question, it could be any of the above. Because it can have multiple possible interpretations, how can there be only one “right” answer to the question. Can they actually tell me I’m wrong when I choose?
This isn’t figuring out a question, it’s gambling that you’ll pick the right assumption.
Why the Question Fails
The question fails because of a number of reasons.
First, and foremost, it provides incomplete and misleading cues. There’s the cup-already-being-full problem.
Second, there is the fact that you simply cannot apply mathematical operations (like addition or subtraction) on nominal or ordinal numbers. For that you must use cardinal or interval numbers.
Third, in order to figure out the answer, you have to already know the answer. On the face of it, the analogy of “coin is to cup, as part is to whole” is not intuitive. Think about this for a second. What possible other container might actually be more appropriate?
Anything? Anything at all? Anything that could give an indication of the relationship between the “part” and the “whole” so that the student can actually do the math?
Even when you break down all the assumptions that have to be made in order to complete the task, and all the other possible assumptions are ignored (the value of the coins, what’s in the cup? Tea? What do coins have to do with a cup of tea? Even if the number 6 is not nominal, is it ounces? Why would you associate the number “6” with the coins instead of the cup?)
By the way, the reason why I chose a hole in the ground instead of a garage for my example is that there is a logical connection between cars and garages, unlike coins and cups. And yet, as we saw, you can force any connection you like if you make the right assumptions. Or wrong ones, as the case may be.
The thing that really pisses me off about Common Core is that it’s not about measuring how much children learn. It’s about proving how smart the test creators are.
OooOOOooohh! Look at the smarty pants! Congratulations! You fooled a 6-year-old! Do you feel like a real man now?!
I’m a huge fan of lateral thinking. However, the issue here is that you are working with six-year-olds. They need to have a foundation of basics before you ask them to think laterally and break the rules.
Mathematics, in fact, is the only discipline where there are actual ‘proofs.’ That means that – when learning the basic fundamentals – a sense of structure makes a great deal of sense. The insanity of the Common Core mathematics questions is that it generates so many debates on the way to interpret the process of answering the questions.
Congratulations: You fooled a 6-year-old! Do you feel like a real man now?!
Why does this matter? Because it is irrational, unreasonable, and illogical to assert that the process by which you make your fundamental assumptions of a problem can be highly interpretive and yet lead to a rigid, definitive, right-or-wrong answer. You simply cannot logically have it both ways.
What’s amazing to me is that the creators of Common Core love complexity to the point of absurdity. It’s as if they have never heard of Occam’s Razor, never even attempted to make sure that this system works. It’s almost like they played a game of MindTrap and said, “Hey, why don’t we build an entire educational system on this!”
This question is the logical end result (or illogical, as the case may be). They were so focused on being clever that they actually got the math wrong. You simply cannot add cardinal numbers and nominal numbers together.
Remember, the actual question that needs to be answered is
6 – 5 = ?
That is the test. That is the equation that students are getting graded on. It’s amazing how much work the creators of Common Core have generated, how difficult they’ve made it, just to get to that point.
I foresee a time when some of these kids are going to enter the work force, get a McJob, and not be able to make basic change. Not because they are stupid, but because the were never taught basic arithmetic. We will think they are stupid, they will blame themselves for being stupid, and the people who are at fault will congratulate themselves for being the cleverest guys in the room.